By using the local dimension-free Harnack inequality established on incomplete Riemannian manifolds, integrability conditions on the coefficients are presented for SDEs to imply the non-explosion of solutions as well as the existence, uniqueness and regularity estimates of invariant probability measures. These conditions include a class of drifts unbounded on compact domains such that the usual Lyapunov conditions can not be verified. The main results are extended to second order differential operators on Hilbert spaces and semi-linear SPDEs.

Non-Gibbsian measures were initially studied in relation to renormalization transformations. In this setting they emerged via phase transitions in the ensemble of original (internal) spins conditioned to a particular renormalized configuration. Later, the same mathematical techniques led to proofs that low-temperature Ising measures subjected to a high-temperature spin-flip evolution can become non-Gibbsian after a finite time. These proofs, however, amount to a "static" view of the evolution as projections of systems with one additional (time) dimension. This was an unsatisfactory state of affairs that did not seem to lead to a truly dynamical understanding of the onset of non-Gibbsianness. As a response to this criticism, in the last few years a new paradigm has been developed in which Gibbs-non-Gibbs transitions are related to changes in the large-deviation rates of conditioned evolutions of measures: Rates with multiple global minima lead to non-Gibbsianness. I will present the main ideas behind this new approach and report on rigorous results for mean-field and local mean-field models.

Several classical lattice models of statistical mechanics, such as percolation and the Ising and Potts models, can be described in terms of clusters. In the last fifteen years, there has been tremendous progress in the study of the geometric properties of such models in two dimensions in the scaling limit, when the lattice spacing is sent to zero. Much of that work has focused on cluster boundaries, using the Schramm-Loewner Evolution (SLE), introduced by Oded Schramm, and collections of SLE loops called Conformal Loop Ensembles (CLEs). In this talk I will discuss the scaling limit of the clusters themselves and their ''areas'' in the case of percolation and the Ising model. This leads to the study of rescaled counting measures and to the concept of Conformal Measure Ensembles (first introduced in joint work with Chuck Newman), with interesting applications to two-dimensional critical percolation and the two-dimensional critical Ising model.

We establish the random k-SAT threshold conjecture for all k exceeding an absolute constant k(0). That is, there is a single critical value c*(k) such that a random k-SAT formula at clause-to-variable ratio c is with high probability satisfiable for c < c*(k), and unsatisfiable for c > c*(k). The threshold c*(k) matches the explicit prediction derived by statistical physicists on the basis of the so-called "one-step replica symmetry breaking" (1RSB) heuristic. In the talk I will describe the main obstacles in computing the threshold, and indicate how they are overcome in our proof.

The minimal spanning tree model has been widely applied to combinatorial optimizations and to the study of disordered physical systems. For the infinite lattice $\mathbb{Z}^d$, rigorous results for the geometry of the minimal spanning forest were recently proved for $d= 2$ and still remain open for $d \geq 3$. We made partial progress by proving that the minimal spanning forest measure is supported on a single tree for quasi-planar graphs, such as the two dimensional slabs. Our proof uses the connections between the minimal spanning forests and critical bond percolations, and certain generalizations of gluing lemmas for bond percolation.